Question

1) Webber Chap. 11 Exercise 1 Prove that {a"b"c"} is not regular. Hint: Copy the proof of Theorem 11.1-only minor alterations are needed.

Answer 1

(I)• Let M=(@, {9, 63, 8, 9 ,F) be any DFA over alphabet {a,b}; st now we will show that han 6" } isn't regular • Griven as for input, ie. L(M) t {an6"} visits a sequence of states: - st (90 E), then s* (q., a), then st(q., aa) & so on since a is finite , M eventually revisits one: -] i & į with icj such that g* (q., ai) = S*Lq., at) • Append 68, le we see that st (qo, al 22 ) = 5*(Qo, ad 6%) So Meither accepts both a fi and as bé, or rejects both anth} contains at bo but not a bt so L(M) 7 {a" by So No DFA has L1M)- 29th A L(M) = {an 6"}-{ try is not regular (II) In this similar way we can also prove that L(m) = {6 cm} = {6 ch} isn't regular for this prove is same as that of (I) part where m= (0, {6, (3, 8,%, I} over alphabet {6, c} (11) So from (I) lele) we have an 69} isn't regular as well as 56 cm} isn't regular So combining both we can easily say that L(M) = { an 6h 3 : { a "6" () isn't regular for M = (a, {9,6, c}, &, qof) be DFA over the alphabet (a, b, c)